The tools I referenced above are mental ones, but you also need a physical tool for cable work. There are a variety of commercial cable needles available and it is really your personal preference that dictates which is the best one. Also, if money is tight or your urge to cable is immediate, you can use a pencil or extra double pointed needle to accomplish a twist or any other cable work. My own preference has led me to the straight, notched cable needle. I find I can manipulate it with greater ease than the hooked variety and the notches really secure my stitches from falling off.
The next thing is to learn what a twisted cable looks like in a charted pattern. As I have mentioned before, it is crucial to study the key of your pattern prior to digging into the knitting. There can be custom symbols for cables, but the ones I have come across tend to follow the same basic notation. For a symmetric twist, the symbol typically looks like the photograph below. These photographs are from an actual pattern. (The paper got a little crinkled because my toddler decided it was page that just had to have some crumpling.) There is both a left and a right twist, so the two you see are representative of each of those and I will include steps for accomplishing both.
In the charts below, the actual twist is illustrated in row one. The subsequent rows have no cabling but are essential to allow the development of your twist. Trying to achieve twisted cables without allowing some spacing between the actual twists would not yield a pleasing result. Always give your twisted cables some room to grow.
Once you get to the cable in your pattern, you will slide the prescribed number of stitches onto your cable needle from the lefthand needle (the stitches in your working row that have not yet been knitted). In my example of three, I slipped three stitches from my left hand needle onto my cable needle, shown in the photo above. I like to hold my working yarn with normal knitting tension and keep it back, out of the way when I do this. It takes a little practice, but gets easier the more you do. The end result will be that your working yarn is still on your righthand needle, the next three stitches will be on your cable needle, and the rest of your unknitted row will remain on your lefthand needle.
For the left twist, you then hold your cable needle in front of your work, meaning it should be between you and your knitting. You then proceed to knit from your lefthand needle like the cable needle is not there. If you find your cable needle getting in your way, just gently hold it still with a free finger as you knit. It is important to be mindful here, though, you do not want any stitches slipping off of your cable needle. It can also feel a little awkward because you will have a little bit of a stretch to make when knitting behind your cable needle since you have essentially skipped stitches that used to be there. In my example, you should knit three stitches from your lefthand needle, the same number of stitches that are being held on the cable needle.
Once you have knit from your lefthand needle, you will pick up your cable needle and, using your righthand needle and working yarn, knit the three stitches off of your cable needle. You simply hold your cable needle in your left hand and treat it like your lefthand knitting needle. Depending what type of cable needle you are using, you might have to manipulate your stitches a little to get them to the point where you can knit them with ease. Obviously, if you are using a hooked needle, you will have to shimmy your stitches out of the u-bend. With notched needles, you will need to free your stitches from the notches so you can knit them on to your righthand needle. Make sure you do not twist your cable needle around. You use to left side of the cable needle to slip the stitches, but you must move the stitches to the right side of the needle before knitting from your cable needle.
I've tried to capture the negative space in the photograph below. Notice how it looks like it is behind the twist? That is the three dimensional affect you want.
Now, as promised, let's look at a right twist. A right twist cable spirals upward from the left to the right, the opposite of the left twist. There is only one crucial difference in the steps to make a right instead of a left twist. Once you slip your stitches onto your cable needle, in our case three stitches, you hold it in the back of your knitting before knitting from your left hand needle This means your knitted fabric and your working yarn will all be in front of your cable needle.
Below, you can see a completed right twist cable. The pattern I am working on frames in the cables nicely with two knitted columns and purled negative space that allows everything to pop forward. The true trick with cables is to take your time and pay attention. Make sure you are careful with the stitches on your cable needle, they will want to wiggle free because they think you are knitting them in the wrong order, the little boogers. Cables are not inherently difficult, but the do take patience. Now go forth and do the twist.